Saturday, 17 March 2012

quantum field theory - photon propagator

Just multiply it out in Fourier, where partial=ik:



bigl[(k2+m2)gmunu+kmuknubigr]fracgnulambda+m2knuklambdak2m2=frac1k2m2bigl((k2+m2)deltamulambda+(k2+m2)m2kmuklambdakmuklambda+kmuk2klambdam2bigr)=deltamulambda



which is a function of k. But converting back to position space, 1(k)=delta(x).



This proves that D is a solution. To be the solution, you usually have to impose boundary conditions, etc. In this case, there are no solutions to bigl[(k2+m2)gmunu+kmuknubigr]fnu=0: the corresponding equation in Fourier is (k2+m2)gmunu+kmuknu=0, and contracting with gmunu gives 0=d(k2+m2)+k2=dm2(d1)k2, where d is the dimension of spacetime, so k2=fracdd1m2, but frac1d1m2gmunu+kmuknu cannot equal 0, as kmuknu cannot be an invertible matrix. So D is the only solution.

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